Higher-order Persistence Diagrams
2026-05-11 • Computational Geometry
Computational Geometry
AI summaryⓘ
The authors propose a new way to analyze topological data by creating 'higher-order persistence diagrams,' which capture relationships between features better than usual methods. Their approach keeps important structural details that other techniques often lose when simplifying the data. They also introduce a faster mathematical method to work with these diagrams without building them fully, making computations much quicker. Testing on random networks shows their method speeds up processing compared to traditional ways.
topological data analysispersistence diagramshigher-order persistencepersistence intervalsharmonic analysiszeta transformfrequency-space evaluationnetwork models
Authors
Charles Fanning, Mehmet Aktas
Abstract
Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for faithful structural comparison and interpretability. We introduce higher-order persistence diagrams, a recursive construction in which containment relations among persistence intervals define higher-order persistence intervals. This construction performs comparison and aggregation directly on persistence diagrams and preserves interval-level structure. We use harmonic analysis to reduce frequency-space evaluations of aggregated diagrams to zeta transforms. This reduction avoids explicit construction of higher-order diagrams and replaces quadratic pair enumeration with nearly linear-time evaluation. Experiments on random network models show substantial speedups over explicit aggregation. Anonymized code is available at https://anonymous.4open.science/r/higher-order-persistence-8201.